Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #129503 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | C2×#346 | C23 |
| prime radical | #62635 | C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ #33326 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,11)(3,15,17,50,20,36,34,53)(5,26)(6,10,23,42)(7,32,24,13)(8,33,25,14)(9,35,37,58,39,18,48,56)(12,44)(16,51,27,41,55,40,57,38)(19,60,21,45,59,52,31,63)(22,61)(28,43)(29,46)(30,47)(49,54)(62,64);(1,3)(2,9)(4,20)(5,27)(6,31)(7,19)(8,34)(10,41)(11,39)(12,45)(13,40)(14,48)(15,49)(16,54)(17,25)(18,29)(21,23)(22,57)(24,59)(26,53)(28,58)(30,55)(32,51)(33,37)(35,62)(36,47)(38,42)(43,63)(44,56)(46,52)(50,61)(60,64);(2,10,11,42)(3,16,17,27,20,55,34,57)(5,28,22,44)(6,7)(8,25)(9,36,48,50,39,15,37,53)(12,46)(13,33,32,14)(18,59,56,31,35,19,58,21)(23,24)(26,61)(29,30,62,54)(38,60,40,63,41,52,51,45)(43,64)Polycyclic presentation:
2^10|a^2=d,c^2=dfh,d^2=g,e^2=hj,f^2=ij,h^2=i,i^2=j,b^a=be,c^a=cf,c^b=ceij,d^b=di,e^a=eh,e^b=ehi,e^c=eh,e^d=ej,f^a=fi,f^b=fij,f^c=fij,g^b=gj,h^a=hi,h^b=hij,h^c=hi,i^a=ij,i^b=ij,i^c=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72227089, #72227141, #72227147, … |
| maximal normal subgroup of | #72227089, #72227141, #72227147, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.4*F.6*F.8);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);